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May 2018 Arbitrarily High Hausdorff Dimensions of Continua
R. Patrick Vernon
Missouri J. Math. Sci. 30(1): 72-76 (May 2018). DOI: 10.35834/mjms/1534384956

Abstract

It is well-known that Hausdorff dimension is not a topological invariant; that is, that two homeomorphic continua can have different Hausdorff dimension, although their topological dimension will be equal. We show that it is possible to take any continuum embeddable in $\mathbb{R}^n$ and embed it in such a way that its Hausdorff dimension is $n$. In doing so, we can obtain an arbitrarily high Hausdorff dimension for any nondegenerate continuum. As an example, we will give different embeddings of an arc whose Hausdorff dimension is any real number between $1$ and $\infty$, including an arc of infinite Hausdorff dimension.

Citation

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R. Patrick Vernon. "Arbitrarily High Hausdorff Dimensions of Continua." Missouri J. Math. Sci. 30 (1) 72 - 76, May 2018. https://doi.org/10.35834/mjms/1534384956

Information

Published: May 2018
First available in Project Euclid: 16 August 2018

zbMATH: 06949051
MathSciNet: MR3844392
Digital Object Identifier: 10.35834/mjms/1534384956

Subjects:
Primary: 28A80
Secondary: 11K55, 54F45

Rights: Copyright © 2018 Central Missouri State University, Department of Mathematics and Computer Science

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Vol.30 • No. 1 • May 2018
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